Surfaces stablement rationnelles sur un corps quasi-fini

TL;DR

The paper investigates conditions under which geometrically rational surfaces over finite fields are not stably rational, focusing on the role of the Brauer group over finite extensions.

Contribution

It establishes that non-rational, smooth, geometrically rational surfaces over finite fields have nonzero Brauer groups over some finite extension, implying they are not stably rational.

Findings

Non-rational surfaces have nonzero Brauer groups over finite extensions

Such surfaces are not stably rational over their base fields

Results apply to geometrically rational surfaces splitting over cyclic extensions

Abstract

If a smooth, geometrically rational surface over a finite field is not rational over that field, then over some finite extension of that field the Brauer group of the surface is nonzero. In particular such a surface is not stably rational. This is a special case of a general statement about geometrically rational surfaces which split over a cyclic extension of their field of definition.